I am somewhat doubting that the lack of grip of the QR nut against the dropout slot is the main problem here.


A toothed lockout of the type used on a Shimano freehub should get plenty of bite by itself. Relying on the nut's grip is insurance, but would also allow at least some forward slippage of the axle and locknut before the play in the QR shaft inside the axle is taken up and begins taking load to assist the toothed locknut.
I think tcpasley gets this. So in this case perhaps the locknut has been reversed or replaced with a smooth one(?).
Viscount hubs often came with brass locknuts that would not grip the driveside dropout slot faces sufficiently, so the driveside locknut needed to be replaced before any hard riding could be done.
QR locknuts often offer ineffective grip, because of uncut "tooth" shaping, lack of "teeth", material choice or lack of any hardening, so in general we should be suspect of their level of additional grip.


So what might be the problem with this bike, if not the slippery fender stay?
I'm supposing it might be an over-extending axle out past the driveside locknut. That, or in combination with a reversed or doubled-up conical spring, the likes of which I have seen in every imaginable form. I had a Peugeot with thin front dropouts almost drop a wheel while I was riding it, simply because the Phil hub's axle stub extension's were relatively too long to accommodate the compressed conical springs, so QR tension was not making it to the critical biting surfaces of the axle end caps (which on these hubs act as the locknuts do to grip the inside dropout faces).


Note that many rear wheels are found to be built with longer axle stub extension on the driveside, meant to accdomodate a claw-mount style derailer hanger, but if/when used on a bike without a bolt-on claw-mount hanger the axle stub with compressed spring will together bottom out inside of the QR nut before sufficient or any real clamping force on the dropout slot faces is realized. It is less common to find these on a wheel with a QR (hollow) axle, but it is not unheard of as this setup did see production.